Problem: Find the least common multiple $(\text{LCM})$ of $12x^4+60x^3+72x^2$ and $135x^3-15x^5$. You can give your answer in its factored form.
Explanation: The least common multiple $(\text{LCM})$ of two polynomial expressions is the polynomial with the least number of factors that is divisible by both polynomials. [How does this relate to the least common multiple of integers?] We can find the $\text{LCM}$ by factoring the two polynomials as much as possible and then comparing the factors: $12x^4+60x^3+72x^2$ can be factored as ${(2^2)}{(3)(x)}{(x)(x+2)}{(x+3)}$ by factoring out a $12x^2$ and using the sum-product pattern. $135x^3-15x^5$ can be factored as ${(5)}{(3)(x^2)}{(x)}{(x+3)}{(3-x)}$ by factoring out a $15x^3$ and using the difference of squares pattern. We can see that: Both polynomials share the factors ${(3)(x)(x+3)}$ Only the first polynomial has the factors ${(2^2)(x)(x+2)}$ Only the second polynomial has the factors ${(5)(x)(3-x)}$ Therefore, the least common multiple is the product of all the above factors: [Why?] $\begin{aligned}&\phantom{=}{(3)(x)(x+3)}{(2^2)(x)(x+2)}{(5)(x)(3-x)}\\\\ &=60(x^3)(x+3)(x+2)(3-x)\end{aligned}$ In conclusion, the least common multiple of the two polynomials is $60(x^3)(x+3)(x+2)(3-x)$.